BCS theory explains conventional superconductivity via phonon-mediated Cooper pairing — but high-Tc cuprates and iron-based superconductors violate BCS assumptions, and the pairing mechanism remains unknown.

The BCS theory (Bardeen, Cooper, Schrieffer 1957) bridges quantum mechanics and materials science to explain conventional superconductivity: phonon-mediated (lattice vibration-mediated) effective electron-electron attraction overcomes Coulomb repulsion to form Cooper pairs — bound states of two electrons with opposite momenta (k↑, −k↓) and spin-singlet configuration. The BCS gap equation Δ = 2ℏω_D exp(−1/N(0)V) gives the superconducting gap, where ω_D is the Debye phonon frequency, N(0) is the density of states at the Fermi level, and V is the effective electron-phonon coupling. The transition temperature T_c ∝ ω_D exp(−1/N(0)V) — explaining why conventional superconductors have low T_c (< 30 K): phonon frequencies are limited. The bridge breaks down for high-T_c superconductors. Cuprate superconductors (Bednorz & Müller 1986, T_c up to 135 K at ambient pressure) and iron-based superconductors have T_c values that violate the BCS ceiling. Their pairing mechanism is the central unsolved problem of condensed matter physics and the primary blocking gap for room-temperature superconductivity (breakthrough gap bg-room-temperature-superconductivity). Candidates include spin-fluctuation mediated pairing, resonating valence bond (RVB) states (Anderson 1987), and polaronic effects — but none is established.

BCS theory requires quantum field theory fluency; materials synthesis of cuprates requires different expertise in solid-state chemistry. The theoretical and experimental communities overlap but insufficiently. High-Tc superconductivity research has become highly specialized, with few researchers spanning ab initio calculations, strongly correlated electron theory, and materials synthesis.